1) is the space of bounded, continuous functions f: X!Y equipped with the uniform metric d 1. Pick xn 2 Kn. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Some "extremal" examples Take any set X and let = {, X}. Let Xbe a compact metric space. A metric space is called sequentially compact if every sequence of elements of has a limit point in . It is separable. Examples. Definition. Hint: Use density of ##\Bbb{Q}## in ##\Bbb{R}##. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. Hence a square is topologically equivalent to a circle, It is definitely complete, because ##\mathbb{R}## is complete. It is sometimes called "rubber-sheet geometry" because the objects can be stretched and contracted like rubber, but cannot be broken. Moreover, a topological space (X,T) is said to be metrizable if there exists a metric for X such that the metric topology T(d) is identical with the topology T. Polish. Topology studies properties of spaces that are invariant under any continuous deformation. Theorem 19. We shall define intuitive topological definitions through it (that will later be converted to the real topological definition), and convert (again, intuitively) calculus definitions of properties (like convergence and continuity) to their topological definition. By Theorem 13, C b(X;Y) is a closed subspace of the complete metric space B(X;Y), so it is a complete metric space. Let X be a metric space and Y a complete metric space. If you are familiar with metric spaces, compare the criteria for the topology τ to the properties of the family of open sets in a metric space: Both the empty set and the whole set are open sets. Any union of open sets is an open set. Proof. Asking that it is closed makes little sense because every topological space … Equivalently: every sequence has a converging sequence. This may be compared with the (ǫ,δ)-definition for a function f: X → Y, from a metric space (X,d) to another metric space (Y,d), to … For example, a square can be deformed into a circle without breaking it, but a figure 8 cannot. Any intersection of finitely many open sets is an open set. y. This distance function is known as the metric. Can you think of a countable dense subset? The topology is closed under arbitrary unions and finite intersections. Proof. 254 Appendix A. A metric space is a special kind of topological space in which there is a distance between any two points. If each Kn 6= ;, then T n Kn 6= ;. The prototype Let X be any metric space and take to be the set of open sets as defined earlier. Example: A bounded closed subset … 1 Metric spaces IB Metric and Topological Spaces 1 Metric spaces 1.1 De nitions As mentioned in the introduction, given a set X, it is often helpful to have a notion of distance between points. Then (C b(X;Y);d 1) is a complete metric space. 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